2 Answers
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It uses a resultant computation. The idea is this. We are given algebraic numbers $x$ and $y$, where $p(x)=0$ and $q(y)=0$ are the minimal polynomials. We want to find the defining polynomial for $z=x+y$. We use $p(x)=p(z-y)$ and $q(y)$, and eliminate $y$ using the classical method of resultants.

This package introduces functions for computation within finite algebraic extensions of rationals. For more information on the notions and algorithms used, see for instance H. Cohen, A Course In Computational Algebraic Number Theory, Springer-Verlag, 1993.

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